The problem of integration in finite terms is the problem of finding exact closed forms for antiderivatives of functions, within a given class of functions. Liouville introduced his elementary functions (built from polynomials, exponentials, logarithms and trigonometric functions) and gave a solution to the problem for that class, nearly 200 years ago. The same problem was shown to be decidable and an algorithm given by Risch in 1969.

We introduce the class of exponentially-algebraic functions, generalising the elementary functions and much more robust than them, and give characterisations of them both in terms of o-minimal local definability and in terms of their types in a reduct of the theory of differentially closed fields.

We then prove the analogue of Liouville's theorem for these exponentially-algebraic functions and give some new decidability results.

This is joint work with Rémi Jaoui, Lyon

Understanding the relationship between expectation and price is central to applications of mathematical finance, including algorithmic trading, derivative pricing and hedging, and the modelling of margin and capital. In this presentation, the link is established via dynamic entropic risk optimisation, which is promoted for its convenient integration into standard pricing methodologies and for its ability to quantify and analyse model risk. As an example of the versatility of entropic pricing, discrete models with classical and quantum information are compared, with studies that demonstrate the effectiveness of quantum decorrelation for model fitting.

We study  dynamic 𝑁-player noncooperative games called 𝛼-potential games, where the change of a player’s objective function upon her unilateral deviation from her strategy is equal to the change of an 𝛼-potential function up to an error 𝛼. Analogous to the static potential game (which corresponds to 𝛼=0), the 𝛼-potential game framework is shown to reduce the challenging task of finding 𝛼-Nash equilibria for a dynamic game to minimizing the 𝛼-potential function. Moreover, an analytical characterization of 𝛼-potential functions is established, with 𝛼 represented in terms of the magnitude of the asymmetry of objective functions’ second-order derivatives. For stochastic differential games in which the state dynamic is a controlled diffusion, 𝛼 is characterized in terms of the number of players, the choice of admissible strategies, and the intensity of interactions and the level of heterogeneity among players. Two classes of stochastic differential games, namely, distributed games and games with mean field interactions, are analyzed to highlight the dependence of 𝛼 on general game characteristics that are beyond the mean field paradigm, which focuses on the limit of 𝑁 with homogeneous players. To analyze the 𝛼-NE (Nash equilibrium), the associated optimization problem is embedded into a conditional McKean–Vlasov control problem. A verification theorem is established to construct 𝛼-NE based on solutions to an infinite-dimensional Hamilton–Jacobi–Bellman equation, which is reduced to a system of ordinary differential equations for linear-quadratic games.